question about martingale

Question

In my lecture notes,I found the following problem:

Let $X$ an $F_{t}$ adapted continuous process and $G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds \right|G_{t}\right)-\int^{t}_{0} E(X_{s}|G_{s})ds $$ is a $G_{t}$ martingale.

Does anybody know how to prove this? Some help would be appreciated

Answer 1

$$ M_{t+h} = E\left( \int^{t+h}_{0}X_{s}ds |G_{t+h}\right) -\int^{t+h}_{0} E(X_{s}|G_{s})ds\\ M_t = E\left( \int^{t}_{0}X_{s}ds| G_{t+h} G_{t}\right)-\int^{t}_{0} E(X_{s}|G_{s})ds \\ M_{t+h} - M_t = E\left( \int^{t+h}_{0}X_{s}ds | G_{t+h}\right) -\int^{t+h}_{t} E(X_{s}|G_{s})ds - E\left( \int^{t}_{0}X_{s}ds | G_{t}\right)\\ E(M_{t+h} - M_t|G_t)= E\left( \int^{t+h}_{0}X_{s}ds | G_{t}\right) -\int^{t+h}_{t} E(X_{s}|G_{t})ds - E\left( \int^{t}_{0}X_{s}ds | G_{t}\right) =0 $$using the linearity of the conditional expectation in the last step.